Graph
layout problems are a class of combinatorial optimization problems whose goal
is to find a layout of an input graph to optimize a certain objective function.
A layout is the embedding of graph G into a host graph H and is defined as a
bijective function mapping the vertices of G to the vertices of H and
associating a path in H for each edge of G. These problems have been shown to
be NP-complete in the general case. Cyclic cutwidth minimization problem and
Vertex bisection minimization are two such problems.

Cyclic
Cutwidth (ccw) Minimization Problem consists of embedding a graph onto a cycle
such that the maximum cut in a region is minimized. Exact results have been
proved for some standard graphs such as complete graphs, complete bipartite
graphs, hypercubes, 2-dimensional cylindrical meshes, 2-dimensional meshes and
partial results have been proved for 3-dimensional meshes.

Using
layout based arguments, we have proved optimal results of cyclic cutwidth for
some classes of graphs such as (m,n)-Tadpole graph, m-book graph, n-sun graph,
cone graph, fan graph, crown graph, web graph, friendship graph, gear graph.
Upper bounds for king graph, join of hypercubes, toroidal mesh, d-dimensional
c-ary cliques, complete split graph and Halin graph have also been proved. We
have also proposed a memetic algorithm for cyclic cutwidth minimization problem
in which we have designed six construction heuristics to generate a good
initial population and a local search operator to improve the solutions in each
generational phase.

Vertex
Bisection Minimization Problem (VBMP) consists of finding a linear layout which
minimizes the number of vertices in the left half of the layout from which
there are edges in the right half of the layout. This problem can also be
defined as partitioning the vertex set V of size n into two sets B and B′
where such that is minimized where λB denotes the vertex width. NP-completeness of
this problem has been proved and also it has been proved that this problem is
polynomially solvable for trees and hypercubes.

We have
proved optimal results for vertex bisection minimization problem for some
classes of graphs such as complete bipartite graph, complete d-ary tree,
threshold graph, complete split graph, m-book graph, friendship graph,
2-dimensional ordinary meshes. We have also designed a branch and bound
algorithm in which we have proposed a strategy to find the lower bound and a
greedy heuristic to find the upper bound in the initial phase.